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Theorem fin1a2lem7 9430
Description: Lemma for fin1a2 9439. Split a III-infinite set in two pieces. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypotheses
Ref Expression
fin1a2lem.b 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
fin1a2lem.aa 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
Assertion
Ref Expression
fin1a2lem7 ((𝐴𝑉 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII)) → 𝐴 ∈ FinIII)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐸
Allowed substitution hints:   𝐴(𝑥)   𝑆(𝑥,𝑦)   𝐸(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem fin1a2lem7
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 peano1 7232 . . . . . 6 ∅ ∈ ω
2 ne0i 4069 . . . . . 6 (∅ ∈ ω → ω ≠ ∅)
3 brwdomn0 8630 . . . . . 6 (ω ≠ ∅ → (ω ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→ω))
41, 2, 3mp2b 10 . . . . 5 (ω ≼* 𝐴 ↔ ∃𝑓 𝑓:𝐴onto→ω)
5 vex 3354 . . . . . . . . . 10 𝑓 ∈ V
6 fof 6256 . . . . . . . . . 10 (𝑓:𝐴onto→ω → 𝑓:𝐴⟶ω)
7 dmfex 7271 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:𝐴⟶ω) → 𝐴 ∈ V)
85, 6, 7sylancr 575 . . . . . . . . 9 (𝑓:𝐴onto→ω → 𝐴 ∈ V)
9 cnvimass 5626 . . . . . . . . . 10 (𝑓 “ ran 𝐸) ⊆ dom 𝑓
10 fdm 6191 . . . . . . . . . . 11 (𝑓:𝐴⟶ω → dom 𝑓 = 𝐴)
116, 10syl 17 . . . . . . . . . 10 (𝑓:𝐴onto→ω → dom 𝑓 = 𝐴)
129, 11syl5sseq 3802 . . . . . . . . 9 (𝑓:𝐴onto→ω → (𝑓 “ ran 𝐸) ⊆ 𝐴)
138, 12sselpwd 4941 . . . . . . . 8 (𝑓:𝐴onto→ω → (𝑓 “ ran 𝐸) ∈ 𝒫 𝐴)
14 fin1a2lem.b . . . . . . . . . . . . . 14 𝐸 = (𝑥 ∈ ω ↦ (2𝑜 ·𝑜 𝑥))
1514fin1a2lem4 9427 . . . . . . . . . . . . 13 𝐸:ω–1-1→ω
16 f1cnv 6301 . . . . . . . . . . . . 13 (𝐸:ω–1-1→ω → 𝐸:ran 𝐸1-1-onto→ω)
17 f1ofo 6285 . . . . . . . . . . . . 13 (𝐸:ran 𝐸1-1-onto→ω → 𝐸:ran 𝐸onto→ω)
1815, 16, 17mp2b 10 . . . . . . . . . . . 12 𝐸:ran 𝐸onto→ω
19 fofun 6257 . . . . . . . . . . . 12 (𝐸:ran 𝐸onto→ω → Fun 𝐸)
2018, 19ax-mp 5 . . . . . . . . . . 11 Fun 𝐸
215resex 5584 . . . . . . . . . . 11 (𝑓 ↾ (𝑓 “ ran 𝐸)) ∈ V
22 cofunexg 7277 . . . . . . . . . . 11 ((Fun 𝐸 ∧ (𝑓 ↾ (𝑓 “ ran 𝐸)) ∈ V) → (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))) ∈ V)
2320, 21, 22mp2an 672 . . . . . . . . . 10 (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))) ∈ V
24 fofun 6257 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → Fun 𝑓)
25 fores 6265 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ (𝑓 “ ran 𝐸) ⊆ dom 𝑓) → (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→(𝑓 “ (𝑓 “ ran 𝐸)))
2624, 9, 25sylancl 574 . . . . . . . . . . . 12 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→(𝑓 “ (𝑓 “ ran 𝐸)))
27 f1f 6241 . . . . . . . . . . . . . . 15 (𝐸:ω–1-1→ω → 𝐸:ω⟶ω)
28 frn 6193 . . . . . . . . . . . . . . 15 (𝐸:ω⟶ω → ran 𝐸 ⊆ ω)
2915, 27, 28mp2b 10 . . . . . . . . . . . . . 14 ran 𝐸 ⊆ ω
30 foimacnv 6295 . . . . . . . . . . . . . 14 ((𝑓:𝐴onto→ω ∧ ran 𝐸 ⊆ ω) → (𝑓 “ (𝑓 “ ran 𝐸)) = ran 𝐸)
3129, 30mpan2 671 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → (𝑓 “ (𝑓 “ ran 𝐸)) = ran 𝐸)
32 foeq3 6254 . . . . . . . . . . . . 13 ((𝑓 “ (𝑓 “ ran 𝐸)) = ran 𝐸 → ((𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→(𝑓 “ (𝑓 “ ran 𝐸)) ↔ (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→ran 𝐸))
3331, 32syl 17 . . . . . . . . . . . 12 (𝑓:𝐴onto→ω → ((𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→(𝑓 “ (𝑓 “ ran 𝐸)) ↔ (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→ran 𝐸))
3426, 33mpbid 222 . . . . . . . . . . 11 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→ran 𝐸)
35 foco 6266 . . . . . . . . . . 11 ((𝐸:ran 𝐸onto→ω ∧ (𝑓 ↾ (𝑓 “ ran 𝐸)):(𝑓 “ ran 𝐸)–onto→ran 𝐸) → (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))):(𝑓 “ ran 𝐸)–onto→ω)
3618, 34, 35sylancr 575 . . . . . . . . . 10 (𝑓:𝐴onto→ω → (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))):(𝑓 “ ran 𝐸)–onto→ω)
37 fowdom 8632 . . . . . . . . . 10 (((𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))) ∈ V ∧ (𝐸 ∘ (𝑓 ↾ (𝑓 “ ran 𝐸))):(𝑓 “ ran 𝐸)–onto→ω) → ω ≼* (𝑓 “ ran 𝐸))
3823, 36, 37sylancr 575 . . . . . . . . 9 (𝑓:𝐴onto→ω → ω ≼* (𝑓 “ ran 𝐸))
395cnvex 7260 . . . . . . . . . . . 12 𝑓 ∈ V
4039imaex 7251 . . . . . . . . . . 11 (𝑓 “ ran 𝐸) ∈ V
41 isfin3-2 9391 . . . . . . . . . . 11 ((𝑓 “ ran 𝐸) ∈ V → ((𝑓 “ ran 𝐸) ∈ FinIII ↔ ¬ ω ≼* (𝑓 “ ran 𝐸)))
4240, 41ax-mp 5 . . . . . . . . . 10 ((𝑓 “ ran 𝐸) ∈ FinIII ↔ ¬ ω ≼* (𝑓 “ ran 𝐸))
4342con2bii 346 . . . . . . . . 9 (ω ≼* (𝑓 “ ran 𝐸) ↔ ¬ (𝑓 “ ran 𝐸) ∈ FinIII)
4438, 43sylib 208 . . . . . . . 8 (𝑓:𝐴onto→ω → ¬ (𝑓 “ ran 𝐸) ∈ FinIII)
45 fin1a2lem.aa . . . . . . . . . . . . . . 15 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4614, 45fin1a2lem6 9429 . . . . . . . . . . . . . 14 (𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸)
47 f1ocnv 6290 . . . . . . . . . . . . . 14 ((𝑆 ↾ ran 𝐸):ran 𝐸1-1-onto→(ω ∖ ran 𝐸) → (𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–1-1-onto→ran 𝐸)
48 f1ofo 6285 . . . . . . . . . . . . . 14 ((𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–1-1-onto→ran 𝐸(𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–onto→ran 𝐸)
4946, 47, 48mp2b 10 . . . . . . . . . . . . 13 (𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–onto→ran 𝐸
50 foco 6266 . . . . . . . . . . . . 13 ((𝐸:ran 𝐸onto→ω ∧ (𝑆 ↾ ran 𝐸):(ω ∖ ran 𝐸)–onto→ran 𝐸) → (𝐸(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω)
5118, 49, 50mp2an 672 . . . . . . . . . . . 12 (𝐸(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω
52 fofun 6257 . . . . . . . . . . . 12 ((𝐸(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω → Fun (𝐸(𝑆 ↾ ran 𝐸)))
5351, 52ax-mp 5 . . . . . . . . . . 11 Fun (𝐸(𝑆 ↾ ran 𝐸))
545resex 5584 . . . . . . . . . . 11 (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))) ∈ V
55 cofunexg 7277 . . . . . . . . . . 11 ((Fun (𝐸(𝑆 ↾ ran 𝐸)) ∧ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))) ∈ V) → ((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))) ∈ V)
5653, 54, 55mp2an 672 . . . . . . . . . 10 ((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))) ∈ V
57 difss 3888 . . . . . . . . . . . . . 14 (𝐴 ∖ (𝑓 “ ran 𝐸)) ⊆ 𝐴
5857, 11syl5sseqr 3803 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → (𝐴 ∖ (𝑓 “ ran 𝐸)) ⊆ dom 𝑓)
59 fores 6265 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ (𝐴 ∖ (𝑓 “ ran 𝐸)) ⊆ dom 𝑓) → (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))))
6024, 58, 59syl2anc 573 . . . . . . . . . . . 12 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))))
61 funcnvcnv 6096 . . . . . . . . . . . . . . . 16 (Fun 𝑓 → Fun 𝑓)
62 imadif 6113 . . . . . . . . . . . . . . . 16 (Fun 𝑓 → (𝑓 “ (ω ∖ ran 𝐸)) = ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸)))
6324, 61, 623syl 18 . . . . . . . . . . . . . . 15 (𝑓:𝐴onto→ω → (𝑓 “ (ω ∖ ran 𝐸)) = ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸)))
6463imaeq2d 5607 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → (𝑓 “ (𝑓 “ (ω ∖ ran 𝐸))) = (𝑓 “ ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸))))
65 difss 3888 . . . . . . . . . . . . . . 15 (ω ∖ ran 𝐸) ⊆ ω
66 foimacnv 6295 . . . . . . . . . . . . . . 15 ((𝑓:𝐴onto→ω ∧ (ω ∖ ran 𝐸) ⊆ ω) → (𝑓 “ (𝑓 “ (ω ∖ ran 𝐸))) = (ω ∖ ran 𝐸))
6765, 66mpan2 671 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → (𝑓 “ (𝑓 “ (ω ∖ ran 𝐸))) = (ω ∖ ran 𝐸))
68 fimacnv 6490 . . . . . . . . . . . . . . . . 17 (𝑓:𝐴⟶ω → (𝑓 “ ω) = 𝐴)
696, 68syl 17 . . . . . . . . . . . . . . . 16 (𝑓:𝐴onto→ω → (𝑓 “ ω) = 𝐴)
7069difeq1d 3878 . . . . . . . . . . . . . . 15 (𝑓:𝐴onto→ω → ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸)) = (𝐴 ∖ (𝑓 “ ran 𝐸)))
7170imaeq2d 5607 . . . . . . . . . . . . . 14 (𝑓:𝐴onto→ω → (𝑓 “ ((𝑓 “ ω) ∖ (𝑓 “ ran 𝐸))) = (𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))))
7264, 67, 713eqtr3rd 2814 . . . . . . . . . . . . 13 (𝑓:𝐴onto→ω → (𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))) = (ω ∖ ran 𝐸))
73 foeq3 6254 . . . . . . . . . . . . 13 ((𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))) = (ω ∖ ran 𝐸) → ((𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))) ↔ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(ω ∖ ran 𝐸)))
7472, 73syl 17 . . . . . . . . . . . 12 (𝑓:𝐴onto→ω → ((𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(𝑓 “ (𝐴 ∖ (𝑓 “ ran 𝐸))) ↔ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(ω ∖ ran 𝐸)))
7560, 74mpbid 222 . . . . . . . . . . 11 (𝑓:𝐴onto→ω → (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(ω ∖ ran 𝐸))
76 foco 6266 . . . . . . . . . . 11 (((𝐸(𝑆 ↾ ran 𝐸)):(ω ∖ ran 𝐸)–onto→ω ∧ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→(ω ∖ ran 𝐸)) → ((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→ω)
7751, 75, 76sylancr 575 . . . . . . . . . 10 (𝑓:𝐴onto→ω → ((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→ω)
78 fowdom 8632 . . . . . . . . . 10 ((((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))) ∈ V ∧ ((𝐸(𝑆 ↾ ran 𝐸)) ∘ (𝑓 ↾ (𝐴 ∖ (𝑓 “ ran 𝐸)))):(𝐴 ∖ (𝑓 “ ran 𝐸))–onto→ω) → ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸)))
7956, 77, 78sylancr 575 . . . . . . . . 9 (𝑓:𝐴onto→ω → ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸)))
80 difexg 4942 . . . . . . . . . . 11 (𝐴 ∈ V → (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ V)
81 isfin3-2 9391 . . . . . . . . . . 11 ((𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ V → ((𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII ↔ ¬ ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸))))
828, 80, 813syl 18 . . . . . . . . . 10 (𝑓:𝐴onto→ω → ((𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII ↔ ¬ ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸))))
8382con2bid 343 . . . . . . . . 9 (𝑓:𝐴onto→ω → (ω ≼* (𝐴 ∖ (𝑓 “ ran 𝐸)) ↔ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII))
8479, 83mpbid 222 . . . . . . . 8 (𝑓:𝐴onto→ω → ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)
85 eleq1 2838 . . . . . . . . . . . 12 (𝑦 = (𝑓 “ ran 𝐸) → (𝑦 ∈ FinIII ↔ (𝑓 “ ran 𝐸) ∈ FinIII))
86 difeq2 3873 . . . . . . . . . . . . 13 (𝑦 = (𝑓 “ ran 𝐸) → (𝐴𝑦) = (𝐴 ∖ (𝑓 “ ran 𝐸)))
8786eleq1d 2835 . . . . . . . . . . . 12 (𝑦 = (𝑓 “ ran 𝐸) → ((𝐴𝑦) ∈ FinIII ↔ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII))
8885, 87orbi12d 902 . . . . . . . . . . 11 (𝑦 = (𝑓 “ ran 𝐸) → ((𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ ((𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)))
8988notbid 307 . . . . . . . . . 10 (𝑦 = (𝑓 “ ran 𝐸) → (¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ ¬ ((𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)))
90 ioran 964 . . . . . . . . . 10 (¬ ((𝑓 “ ran 𝐸) ∈ FinIII ∨ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII) ↔ (¬ (𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII))
9189, 90syl6bb 276 . . . . . . . . 9 (𝑦 = (𝑓 “ ran 𝐸) → (¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ (¬ (𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)))
9291rspcev 3460 . . . . . . . 8 (((𝑓 “ ran 𝐸) ∈ 𝒫 𝐴 ∧ (¬ (𝑓 “ ran 𝐸) ∈ FinIII ∧ ¬ (𝐴 ∖ (𝑓 “ ran 𝐸)) ∈ FinIII)) → ∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9313, 44, 84, 92syl12anc 1474 . . . . . . 7 (𝑓:𝐴onto→ω → ∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
94 rexnal 3143 . . . . . . 7 (∃𝑦 ∈ 𝒫 𝐴 ¬ (𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) ↔ ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9593, 94sylib 208 . . . . . 6 (𝑓:𝐴onto→ω → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9695exlimiv 2010 . . . . 5 (∃𝑓 𝑓:𝐴onto→ω → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
974, 96sylbi 207 . . . 4 (ω ≼* 𝐴 → ¬ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII))
9897con2i 136 . . 3 (∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) → ¬ ω ≼* 𝐴)
99 isfin3-2 9391 . . 3 (𝐴𝑉 → (𝐴 ∈ FinIII ↔ ¬ ω ≼* 𝐴))
10098, 99syl5ibr 236 . 2 (𝐴𝑉 → (∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII) → 𝐴 ∈ FinIII))
101100imp 393 1 ((𝐴𝑉 ∧ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ FinIII ∨ (𝐴𝑦) ∈ FinIII)) → 𝐴 ∈ FinIII)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 834   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  wss 3723  c0 4063  𝒫 cpw 4297   class class class wbr 4786  cmpt 4863  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  cima 5252  ccom 5253  Oncon0 5866  suc csuc 5868  Fun wfun 6025  wf 6027  1-1wf1 6028  ontowfo 6029  1-1-ontowf1o 6030  (class class class)co 6793  ωcom 7212  2𝑜c2o 7707   ·𝑜 comu 7711  * cwdom 8618  FinIIIcfin3 9305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seqom 7696  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-wdom 8620  df-card 8965  df-fin4 9311  df-fin3 9312
This theorem is referenced by:  fin1a2lem8  9431
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